Spinorial q-number anti-commutator function
for Weyl fields in quantum Einstein gravity
Ritsu Yoshida
∗Abstract
A gravitational version of the S function for 2-component massless spinor fields or Weyl fields is introduced in the manifestly covariant operator formalism of quantum gravity. This function is defined as a bilocal operator that relates to the 4D anti-commutation relation between a Weyl field and its Hermitian conjugate. Some properties of this function are investigated.
1.
Introduction
The 4D anti-commutation relation between a free Dirac field ψ(x) with mass m and its Dirac conjugate ¯ψ(y) is given by
{ψ(x), ¯ψ(y)} = iS(x − y; m) , (1.1) in the Minkowski spacetime [1]. Here, S(x− y; m) is the S function for
Dirac fields defined by
(iγμ∂μx− m)S(x − y; m) = 0 , (1.2) S(x− y; m)|x0=y0 =−iγ0δ3(x− y) , (1.3)
with the use of the ordinary gamma matrices γμ. Abe and Nakanishi [2] introduced an extended version of the S function in their method for solving
with ◦ σab≡ 1 4( ◦ γa ◦ γb− ◦ γb ◦ γa) , (1.11) ωab μ ≡ 1 2[h ρa(∂ μhρb− ∂ρhμb)− hρb(∂μhρa− ∂ρhμa) +hμchρahσb(∂σhρc− ∂ρhσc)] . (1.12) Here ωab
μ denotes the spin connection [1].
If m = 0, then we can decompose Dirac fields into two 2-component spinor fields. These are called Weyl fields [4]. So, we need to treat Weyl fields in the vierbein formalism of quantum gravity, and consequently to have a quantum-gravity version of the S function for Weyl fields.
The purpose of the present paper is to introduce the quantum-gravity S function for Weyl fields and to investigate it. For this purpose, we analo-gously use the properties of the quantum-gravity S function for Dirac fields defined by Eqs. (1.6) and (1.7).
This paper is organized as follows. In the next section, we briefly review the manifestly covariant operator formalism for Weyl fields interacting with the gravitational field. In Sect. 3, we introduce the quantum-gravity S function for Weyl fields. In Sect. 4, we investigate its transformation properties. The last section is devoted to discussion.
2.
Covariant operator formalism for Weyl fields
We consider a quantum system in which Weyl fields interact with the grav-itational field. Let us call it the quantum coupled Einstein–Weyl system.
2.1. Lagrangian density for Weyl fields
In order to treat Weyl fields in this system, we extract a corresponding Lagrangian density from the following one [1] for a massless Dirac field the manifestly covariant operator formalism of quantum electrodynamics.
In this method, the electromagnetic field and the electron one are expanded in powers of e2 where e denotes the electromagnetic coupling constant.
The zeroth-order of the S function, S(0)(x, y), is defined by the following
q-number Cauchy problem:
{iγμ[∂μx− iA(0)μ (x)]− m}S(0)(x, y) = 0 , (1.4) S(0)(x, y)
|x0=y0 =−iγ0δ3(x− y) . (1.5)
Here A(0)μ (x) is the zeroth order of the electromagnetic field.
Comparing Eqs. (1.2) and (1.3) with Eqs. (1.4) and (1.5), and taking account of the vierbein formalism of quantum gravity [1], we defined a quantum-gravity version of the S function S(x, y; m) [3] for Dirac fields
interacting with the gravitational field hμa(x) (a = 0, 1, 2, 3). Namely, we form the following q-number Cauchy problem:
ih(x)γμ(x)[∂μx+ ωμ(x)]S(x, y; m) − mh(x)S(x, y; m) = 0 , (1.6) S(x, y; m)|x0=y0 =−i γ 0(x) h(x)g00(x)δ 3(x − y) . (1.7)
Here, h = det hμa, γμ ≡ hμa ◦γa with the flat-space gamma matrices ◦ γa (a = 0, 1, 2, 3), {γ◦a, ◦ γb} = 2ηab, ηab= diag(+1,−1, −1, −1) , (1.8) and gμν = ηabhμahνb. (1.9)
As in Ref. [3], we use Greek small letters for GL(4) indexes and italic small letters for internal Lorentz ones. The symbol ωμ in (1.6) is defined by
ωμ≡ 1 2ω ab μ ◦ σab, (1.10)
with ◦ σab≡1 4( ◦ γa ◦ γb− ◦ γb ◦ γa) , (1.11) ωab μ ≡ 1 2[h ρa(∂ μhρb− ∂ρhμb)− hρb(∂μhρa− ∂ρhμa) +hμchρahσb(∂σhρc− ∂ρhσc)] . (1.12) Here ωab
μ denotes the spin connection [1].
If m = 0, then we can decompose Dirac fields into two 2-component spinor fields. These are called Weyl fields [4]. So, we need to treat Weyl fields in the vierbein formalism of quantum gravity, and consequently to have a quantum-gravity version of the S function for Weyl fields.
The purpose of the present paper is to introduce the quantum-gravity S function for Weyl fields and to investigate it. For this purpose, we analo-gously use the properties of the quantum-gravity S function for Dirac fields defined by Eqs. (1.6) and (1.7).
This paper is organized as follows. In the next section, we briefly review the manifestly covariant operator formalism for Weyl fields interacting with the gravitational field. In Sect. 3, we introduce the quantum-gravity S function for Weyl fields. In Sect. 4, we investigate its transformation properties. The last section is devoted to discussion.
2.
Covariant operator formalism for Weyl fields
We consider a quantum system in which Weyl fields interact with the grav-itational field. Let us call it the quantum coupled Einstein–Weyl system.
2.1. Lagrangian density for Weyl fields
In order to treat Weyl fields in this system, we extract a corresponding Lagrangian density from the following one [1] for a massless Dirac field the manifestly covariant operator formalism of quantum electrodynamics.
In this method, the electromagnetic field and the electron one are expanded in powers of e2 where e denotes the electromagnetic coupling constant.
The zeroth-order of the S function, S(0)(x, y), is defined by the following
q-number Cauchy problem:
{iγμ[∂xμ− iA(0)μ (x)]− m}S(0)(x, y) = 0 , (1.4) S(0)(x, y)
|x0=y0=−iγ0δ3(x− y) . (1.5)
Here A(0)μ (x) is the zeroth order of the electromagnetic field.
Comparing Eqs. (1.2) and (1.3) with Eqs. (1.4) and (1.5), and taking account of the vierbein formalism of quantum gravity [1], we defined a quantum-gravity version of the S function S(x, y; m) [3] for Dirac fields
interacting with the gravitational field hμa(x) (a = 0, 1, 2, 3). Namely, we form the following q-number Cauchy problem:
ih(x)γμ(x)[∂μx+ ωμ(x)]S(x, y; m) − mh(x)S(x, y; m) = 0 , (1.6) S(x, y; m)|x0=y0=−i γ 0(x) h(x)g00(x)δ 3(x − y) . (1.7)
Here, h = det hμa, γμ ≡ hμa ◦γa with the flat-space gamma matrices ◦ γa (a = 0, 1, 2, 3), {γ◦a, ◦ γb} = 2ηab, ηab= diag(+1,−1, −1, −1) , (1.8) and gμν = ηabhμahνb. (1.9)
As in Ref. [3], we use Greek small letters for GL(4) indexes and italic small letters for internal Lorentz ones. The symbol ωμ in (1.6) is defined by
ωμ ≡ 1 2ω ab μ ◦ σab, (1.10)
Combining (1.11) and (2.3), we write ◦ σab= ⎛ ⎝ ( ◦ Sab)AB 0 0 ( ◦ ¯ Sab)A˙B˙ ⎞ ⎠ , (2.10) with (S◦ab)AB ≡ 1 4( ◦ σa ◦ ¯ σb− ◦ σb ◦ ¯ σa)AB, (2.11) ( ◦ Sab) ˙ A ˙ B ≡ 1 4( ◦ ¯ σa ◦ σb− ◦ ¯ σb ◦ σa) ˙ A ˙ B=−[( ◦ Sab)†] ˙ A ˙ B. (2.12)
Equations (2.8), (2.9), (2.11), and (2.12) yield (σ◦a ◦ ¯ Sbc− ◦ Sbcσ◦a)A ˙B = ηab(σ◦c)A ˙B− ηac(σ◦b)A ˙B, (2.13) (σ¯◦a ◦ Sbc− ◦ ¯ Sbc ◦ ¯ σa) ˙ AB = η ab( ◦ ¯ σc) ˙ AB − ηac( ◦ ¯ σb) ˙ AB, (2.14) and (S◦ab ◦ Scd− ◦ Scd ◦ Sab)AB = ηad( ◦ Sbc)AB− ηac( ◦ Sbd)AB+ ηbc( ◦ Sad)AB− ηbd( ◦ Sac)AB, (2.15) ( ◦ ¯ Sab ◦ ¯ Scd− ◦ ¯ Scd ◦ ¯ Sab) ˙ A ˙ B = ηad( ◦ ¯ Sbc) ˙ A ˙ B− ηac( ◦ ¯ Sbd) ˙ A ˙ B+ ηbc( ◦ ¯ Sad) ˙ A ˙ B− ηbd( ◦ ¯ Sac) ˙ A ˙ B. (2.16) Inserting Eqs. (2.2) and (2.3) to Eq. (2.1), we obtain the sum of two Lagrangian densities: LD=Lξ+Lη, (2.17) where Lξ≡ i 2[hξ †hμa ◦¯σ a(∂μ+ ωμ)ξ− ξ†(←∂−μ− ωμ)hμa ◦σ¯a ξh] , (2.18) Lη≡ i 2[hη †hμa ◦σ a (∂μ+ ωμ)η− η†(←∂−μ− ωμ)hμa ◦σaηh] . (2.19) ψ(x) interacting with the gravitational field:
LD= i 2[h ¯ψγ μ(∂ μ+ ωμ)ψ− ¯ψ(←∂−μ− ωμ)γμψh] , (2.1) where ¯ψ≡ ψ† ◦γ 0.
Massless Dirac fields are equivalent to pairs of two corresponding Weyl fields. Therefore, we express ψ(x) andγ◦a as follows [4]:
ψ(x)≡ ⎛ ⎝ ξA(x) ηB˙(x) ⎞ ⎠ , (2.2) and ◦ γa≡ ⎛ ⎝ 0 ( ◦ σa)A ˙B (σ¯◦a)AB˙ 0 ⎞ ⎠ . (2.3)
Here, ξAand ηB˙ are Weyl fields,σ◦a and ◦ ¯
σa consist of the unit matrix and the Pauli spin matrices,
(σ◦a)A ˙B ≡ (σ0, σ1, σ2, σ3)A ˙B , (2.4) (σ¯◦a)
˙
AB
≡ (σ0,−σ1,−σ2,−σ3)A ˙B . (2.5)
We use capital Roman letters, A, B, C, etc., for 2-component spinor in-dexes, and raise or lower them by
�AB= �AB ≡ ⎛ ⎝ 0 1 −1 0 ⎞ ⎠ . (2.6)
Of course,σ◦¯a is a complex conjugate of ◦ σa: (σ◦¯a) ˙ AB=�(σ◦ a)A ˙B�∗, (2.7) with (2.6). Inserting (2.3) to (1.8), we obtain (σ◦a ◦ ¯ σb+ ◦ σb ◦ ¯ σa)AB = 2ηabδAB, (2.8) (¯σ◦a ◦ σb+ ◦ ¯ σb ◦ σa) ˙ A ˙ B = 2ηabδ ˙ A ˙ B. (2.9)
Combining (1.11) and (2.3), we write ◦ σab= ⎛ ⎝ ( ◦ Sab)AB 0 0 ( ◦ ¯ Sab)A˙B˙ ⎞ ⎠ , (2.10) with (S◦ab)AB≡ 1 4( ◦ σa ◦ ¯ σb− ◦ σb ◦ ¯ σa)AB, (2.11) ( ◦ Sab) ˙ A ˙ B≡ 1 4( ◦ ¯ σa ◦ σb− ◦ ¯ σb ◦ σa) ˙ A ˙ B =−[( ◦ Sab)†] ˙ A ˙ B. (2.12)
Equations (2.8), (2.9), (2.11), and (2.12) yield (σ◦a ◦ ¯ Sbc− ◦ Sbcσ◦a)A ˙B= ηab(σ◦c)A ˙B− ηac(σ◦b)A ˙B, (2.13) (σ◦¯a ◦ Sbc− ◦ ¯ Sbc ◦ ¯ σa) ˙ AB= η ab( ◦ ¯ σc) ˙ AB − ηac( ◦ ¯ σb) ˙ AB, (2.14) and (S◦ab ◦ Scd− ◦ Scd ◦ Sab)AB = ηad( ◦ Sbc)AB− ηac( ◦ Sbd)AB+ ηbc( ◦ Sad)AB− ηbd( ◦ Sac)AB, (2.15) ( ◦ ¯ Sab ◦ ¯ Scd− ◦ ¯ Scd ◦ ¯ Sab) ˙ A ˙ B = ηad( ◦ ¯ Sbc) ˙ A ˙ B− ηac( ◦ ¯ Sbd) ˙ A ˙ B+ ηbc( ◦ ¯ Sad) ˙ A ˙ B− ηbd( ◦ ¯ Sac) ˙ A ˙ B. (2.16) Inserting Eqs. (2.2) and (2.3) to Eq. (2.1), we obtain the sum of two Lagrangian densities: LD=Lξ+Lη, (2.17) where Lξ ≡ i 2[hξ †hμa ◦σ¯ a (∂μ+ ωμ)ξ− ξ†(←∂−μ− ωμ)hμa ◦σ¯aξh] , (2.18) Lη ≡ i 2[hη †hμa ◦σ a(∂μ+ ωμ)η− η†(←∂−μ− ωμ)hμa ◦σaηh] . (2.19) ψ(x) interacting with the gravitational field:
LD= i 2[h ¯ψγ μ(∂ μ+ ωμ)ψ− ¯ψ(←∂−μ− ωμ)γμψh] , (2.1) where ¯ψ≡ ψ† ◦γ 0.
Massless Dirac fields are equivalent to pairs of two corresponding Weyl fields. Therefore, we express ψ(x) andγ◦a as follows [4]:
ψ(x)≡ ⎛ ⎝ ξA(x) ηB˙(x) ⎞ ⎠ , (2.2) and ◦ γa≡ ⎛ ⎝ 0 ( ◦ σa)A ˙B (σ¯◦a)AB˙ 0 ⎞ ⎠ . (2.3)
Here, ξA and ηB˙ are Weyl fields,σ◦a and ◦ ¯
σa consist of the unit matrix and the Pauli spin matrices,
(σ◦a)A ˙B ≡ (σ0, σ1, σ2, σ3)A ˙B , (2.4) (σ¯◦a)
˙
AB
≡ (σ0,−σ1,−σ2,−σ3)A ˙B . (2.5)
We use capital Roman letters, A, B, C, etc., for 2-component spinor in-dexes, and raise or lower them by
�AB= �AB≡ ⎛ ⎝ 0 1 −1 0 ⎞ ⎠ . (2.6)
Of course, ¯σ◦a is a complex conjugate of ◦ σa: (σ¯◦a) ˙ AB=�(σ◦ a)A ˙B�∗, (2.7) with (2.6). Inserting (2.3) to (1.8), we obtain (σ◦a ◦ ¯ σb+ ◦ σb ◦ ¯ σa)AB= 2ηabδAB, (2.8) (σ¯◦a ◦ σb+ ◦ ¯ σb ◦ σa) ˙ A ˙ B= 2ηabδ ˙ A ˙ B. (2.9)
where δ3 denotes the spatial delta function Π3
k=1δ(xk− yk). In the above and hereafter, a prime attached to a spacetime function means that its argument is not xλ but yλ where it is understood x0 = y0. Using the
field equation (2.25), the canonical conjugate (2.27), and the equal-time canonical anti-commutation relation (2.28), we obtain
{ξA, (ξ†)�B˙} =
h0a
hg00(
◦
σa)A ˙Bδ3. (2.29)
The action integral of ˜LW in (2.22) is invariant under the gravitational BRST transformation. The corresponding charge [1] is defined by
QG≡
d3x hg0ν(bρ∂νcρ− ∂νbρ· cρ) , (2.30) where bρ is the gravitational B-field and cρ is the gravitational Faddeev– Popov ghost field. The anti-commutation relation between QG and the
Weyl field ξ is given by
{iQG , ξ} = i
d3y h�g00�[ ˙b�ρ, ξ ] cρ�. (2.31) Inserting the commutator,
[ ˙b�ρ, ξ ] =− iκ hg00∂ρξ· δ
3, (2.32)
to the right-hand side of (2.31), we obtain the gravitational BRST transfor-mation of ξ. Here, κ is Einstein’s gravitational constant. The commutator (2.32) is derived from the combination of (2.26) and the following commu-tators [1]: [ hμa, b� ρ] = iκ hg00h 0aδμ ρδ3, (2.33) [ gμν, b�ρ] = iκ hg00(g μ0δν ρ+ gν0δμρ)δ3, (2.34) [ ωμab, b�ρ] =− iκ hg00δ 0 μωρabδ3. (2.35)
The symbols ωμ and ωμ are defined by (ωμ)AB ≡ 1 2ω ab μ ( ◦ Sab)AB, (2.20) (ωμ) ˙ A ˙ B≡ 1 2ω ab μ ( ◦ Sab) ˙ A ˙ B , (2.21) respectively.
2.2. Properties of field operators
Now we useLξ in (2.18) because it is enough to treat one of the pair of ξ and η. In order to take ξ as the canonical variable, we replaceLξ by
˜
LW ≡ ihξ†hμa ◦¯σa(∂μ+ ωμ)ξ . (2.22) The discarded term is a total divergence,
Lξ− ˜LW =−i 2∂μ(hξ
†hμa ◦σ¯
a ξ) , (2.23)
with the use of
∂μ(hhμa)· ◦ ¯ σa= hhμa( ◦ ¯ σa ωμ− ωμ ◦ ¯ σa) . (2.24)
The field equation,
ihhμa ◦σ¯a(∂μ+ ωμ)ξ = 0 , (2.25) is derived from ˜LW. Modifying this equation, we see
˙ξ = −hg0a00
◦
σa(hkb ◦σ¯b ∂k+ hμb ◦σ¯bωμ)ξ . (2.26) The canonical conjugate of ξ is defined by
πξA≡ ∂ ˜LW ∂ ˙ξA =−ih(ξ†)B˙h0a( ◦ ¯ σa) ˙ BA, (2.27)
via the “left” functional derivative with respect to ξ. The equal-time canon-ical anti-commutation relation is set as follows:
where δ3 denotes the spatial delta function Π3
k=1δ(xk− yk). In the above and hereafter, a prime attached to a spacetime function means that its argument is not xλ but yλ where it is understood x0 = y0. Using the
field equation (2.25), the canonical conjugate (2.27), and the equal-time canonical anti-commutation relation (2.28), we obtain
{ξA, (ξ†)�B˙} =
h0a
hg00(
◦
σa)A ˙Bδ3. (2.29)
The action integral of ˜LW in (2.22) is invariant under the gravitational BRST transformation. The corresponding charge [1] is defined by
QG≡
d3x hg0ν(bρ∂νcρ− ∂νbρ· cρ) , (2.30) where bρ is the gravitational B-field and cρ is the gravitational Faddeev– Popov ghost field. The anti-commutation relation between QG and the
Weyl field ξ is given by
{iQG , ξ} = i
d3y h�g00�[ ˙b�ρ, ξ ] cρ�. (2.31) Inserting the commutator,
[ ˙b�ρ, ξ ] =− iκ hg00∂ρξ· δ
3, (2.32)
to the right-hand side of (2.31), we obtain the gravitational BRST transfor-mation of ξ. Here, κ is Einstein’s gravitational constant. The commutator (2.32) is derived from the combination of (2.26) and the following commu-tators [1]: [ hμa, b� ρ] = iκ hg00h 0aδμ ρδ3, (2.33) [ gμν, b�ρ] = iκ hg00(g μ0δν ρ+ gν0δμρ)δ3, (2.34) [ ωabμ , b�ρ] =− iκ hg00δ 0 μωabρ δ3. (2.35)
The symbols ωμand ωμ are defined by (ωμ)AB ≡ 1 2ω ab μ ( ◦ Sab)AB, (2.20) (ωμ) ˙ A ˙ B≡ 1 2ω ab μ ( ◦ Sab) ˙ A ˙ B , (2.21) respectively.
2.2. Properties of field operators
Now we use Lξ in (2.18) because it is enough to treat one of the pair of ξ and η. In order to take ξ as the canonical variable, we replaceLξ by
˜
LW ≡ ihξ†hμa ◦σ¯a (∂μ+ ωμ)ξ . (2.22) The discarded term is a total divergence,
Lξ− ˜LW =−i 2∂μ(hξ
†hμa ◦σ¯
a ξ) , (2.23)
with the use of
∂μ(hhμa)· ◦ ¯ σa= hhμa( ◦ ¯ σaωμ− ωμ ◦ ¯ σa) . (2.24)
The field equation,
ihhμa ◦σ¯a(∂μ+ ωμ)ξ = 0 , (2.25) is derived from ˜LW. Modifying this equation, we see
˙ξ = −hg0a00
◦
σa(hkb ◦σ¯b∂k+ hμb ◦σ¯bωμ)ξ . (2.26) The canonical conjugate of ξ is defined by
πξA≡ ∂ ˜LW ∂ ˙ξA =−ih(ξ†)B˙h0a( ◦ ¯ σa) ˙ BA, (2.27)
via the “left” functional derivative with respect to ξ. The equal-time canon-ical anti-commutation relation is set as follows:
(1.7) for S(x, y; m); Eq. (1.6) relates to the gravitational Dirac equation, and Eq. (1.7) is the quantum-gravity version of the condition (1.3) for the ordinary S function.
Let us have two equations to define the quantum-gravity S function
SA ˙B(x, y) for ξ. Comparing the field equation (2.25) for ξ(x) with the gravitational Dirac equation for ψ(x) from LD in (2.1), we firstly have a
form of ih(x)hμa(x)(σ◦¯ a) ˙ CD {δ A D ∂xμ+ [ωμ(x)]DA}SA ˙B(x, y) = 0 . (3.1) Comparing the equal-time anti-commutation relation (2.29) with that be-tween ψ(x) and ¯ψ(y) [3], we secondly have a form of
SA ˙B(x, y)|0=−i
h0a(x)
h(x)g00(x)(
◦
σa)A ˙Bδ3, (3.2) where the symbol|0denotes to set x0= y0. Using the set of Eqs. (3.1) and
(3.2), we form the q-number Cauchy problem for the quantum-gravity S functionSA ˙B(x, y). This function is a bilocal operator and does not depend upon x− y alone, because of the same reason for the quantum-gravity D functionD(x, y) [5].
We regard the Hermitian conjugate ofSA ˙B(x, y) as follows:
[SA ˙B(x, y)]†=−SB ˙A(y, x) . (3.3) This is analogous to that of the ordinary S function for Dirac fields [3]. Thus, we have SA ˙B(x, y){←∂−μyδ ˙ B ˙ C− [ωμ(y)] ˙ B ˙ C}( ◦ ¯ σa) ˙ CDhμa(y)h(y)i = 0 . (3.4) Equations (3.1) and (3.4) denote that the index A ofSA ˙B(x, y) relates to the spinor property at the poinit x while the index ˙B does to that at the
point y. In addition, the Lagrangian density ˜LW is invariant under the internal
Lorentz BRST transformation. The corresponding charge [1] is defined by
QL≡
d3xhg0ν[sab(Dνt)ab− ∂νsab· tab+ i∂ν¯tab· tbctac] , (2.36) with
(Dμt)ab≡ ∂μtab+ ωacμ tcb− ωμbctca. (2.37) Here, sab is the internal Lorentz B-field, and tab and ¯tab are the internal Lorentz Faddeev–Popov ghost and ghost fields; these fields are anti-symmetric with respect to the indexes a and b. The anti-commutation relation between QL and ξ is given by
{iQL, ξ} = i
d3y h�g00�[ ˙s�ab, ξ ] tab�. (2.38) Inserting the commutator,
[ ˙s�ab, ξ ] =− i
2hg00
◦
Sabξ δ3, (2.39)
to the right-hand side of (2.38), we obtain the internal Lorentz BRST trans-formation of ξ. The commutator (2.39) is derived from the combination of (2.26) and the following commutator [1]:
[ ωμab, s�cd] = iδ0 μ 2hg00(δ a cδbd− δbcδad)δ3. (2.40) The BRST charges, QG and QL, give us the subsidiary conditions,
QG|phys� = 0 , QL|phys� = 0 , (2.41)
to define the physical subspace of the indefinite-metric Hilbert space.
3.
Quantum-gravity S function for Weyl fields
In order to define the S function for Weyl fields, we form a q-number Cauchy problem for it. This is analogous to the set of Eqs. (1.6) and
(1.7) forS(x, y; m); Eq. (1.6) relates to the gravitational Dirac equation, and Eq. (1.7) is the quantum-gravity version of the condition (1.3) for the ordinary S function.
Let us have two equations to define the quantum-gravity S function
SA ˙B(x, y) for ξ. Comparing the field equation (2.25) for ξ(x) with the gravitational Dirac equation for ψ(x) from LD in (2.1), we firstly have a
form of ih(x)hμa(x)(σ◦¯ a) ˙ CD {δ A D ∂μx+ [ωμ(x)]DA}SA ˙B(x, y) = 0 . (3.1) Comparing the equal-time anti-commutation relation (2.29) with that be-tween ψ(x) and ¯ψ(y) [3], we secondly have a form of
SA ˙B(x, y)|0=−i
h0a(x)
h(x)g00(x)(
◦
σa)A ˙Bδ3, (3.2) where the symbol|0denotes to set x0= y0. Using the set of Eqs. (3.1) and
(3.2), we form the q-number Cauchy problem for the quantum-gravity S functionSA ˙B(x, y). This function is a bilocal operator and does not depend upon x− y alone, because of the same reason for the quantum-gravity D functionD(x, y) [5].
We regard the Hermitian conjugate ofSA ˙B(x, y) as follows:
[SA ˙B(x, y)]†=−SB ˙A(y, x) . (3.3) This is analogous to that of the ordinary S function for Dirac fields [3]. Thus, we have SA ˙B(x, y){←∂−μyδ ˙ B ˙ C− [ωμ(y)] ˙ B ˙ C}( ◦ ¯ σa) ˙ CDhμa(y)h(y)i = 0 . (3.4) Equations (3.1) and (3.4) denote that the index A of SA ˙B(x, y) relates to the spinor property at the poinit x while the index ˙B does to that at the
point y. In addition, the Lagrangian density ˜LW is invariant under the internal
Lorentz BRST transformation. The corresponding charge [1] is defined by
QL≡
d3xhg0ν[sab(Dνt)ab− ∂νsab· tab+ i∂ν¯tab· tbctac] , (2.36) with
(Dμt)ab≡ ∂μtab+ ωμactcb− ωbcμ tca. (2.37) Here, sab is the internal Lorentz B-field, and tab and ¯tab are the internal Lorentz Faddeev–Popov ghost and ghost fields; these fields are anti-symmetric with respect to the indexes a and b. The anti-commutation relation between QL and ξ is given by
{iQL, ξ} = i
d3y h�g00�[ ˙s�ab, ξ ] tab�. (2.38) Inserting the commutator,
[ ˙s�ab, ξ ] =− i
2hg00
◦
Sabξ δ3, (2.39)
to the right-hand side of (2.38), we obtain the internal Lorentz BRST trans-formation of ξ. The commutator (2.39) is derived from the combination of (2.26) and the following commutator [1]:
[ ωabμ , s�cd] = iδ0 μ 2hg00(δ a cδbd− δbcδad)δ3. (2.40) The BRST charges, QG and QL, give us the subsidiary conditions,
QG|phys� = 0 , QL|phys� = 0 , (2.41)
to define the physical subspace of the indefinite-metric Hilbert space.
3.
Quantum-gravity S function for Weyl fields
In order to define the S function for Weyl fields, we form a q-number Cauchy problem for it. This is analogous to the set of Eqs. (1.6) and
with
KA ˙B λ
(x, y, z)≡ i SA ˙C(x, y)h(y)hλa(y)( ◦ ¯ σa) ˙ CD SD ˙B(y, z) . (3.11) The nonlocal “current” KA ˙Bλ(x, y, z) is conserved with respect to y by virtue of Eqs. (2.24), (3.1), and (3.4). Of course, the right-hand side of (3.10) is independent of y0and reduces to
SA ˙B(x, z) by setting y0= x0via (3.2).
Using Eqs. (2.29) and (3.7), we obtain the 4D anti-commutation rela-tion between ξA(x) and (ξ†)B˙(y) as follows:
{ξA(x), (ξ†)B˙(y)} = i SA ˙B(x, y) +RA ˙B(x, y; ξ†) , (3.12)
whereRA ˙B(x, y; ξ†) contains a commutator betweenSA ˙B(x, y) and (ξ†)C˙(z).
4.
Transformation properties
In the quantum coupled Einstein–Weyl system, there are the affine, the gravitational BRST, and the internal Lorentz BRST symmetries. We in-vestigate the transformation properties of the quantum-gravity S function
SA ˙B(x, y) with respect to these three symmetries.
4.1. Affine transformation
Let ˆPλand ˆMκλbe the translation generator and the GL(4) one [1], respec-tively. In what follows, we show that the affine transformation ofSA ˙B(x, y) is given by [ i ˆPλ,SA ˙B(x, y) ] = (∂λx+ ∂ y λ)SA ˙B(x, y) , (4.1) [ i ˆMκλ,SA ˙B(x, y) ] = (x κ∂x λ+ yκ∂ y λ)SA ˙B(x, y) . (4.2) As in Ref. [3], we prove only (4.2) since ˆPλ can formally be regarded as
ˆ
M5
λ with x5= y5= 1, and δμ5= δ5ν= 0. Modifying Eqs. (3.1) and (3.4), we see
∂x 0SA ˙B(x, y)|0= i h0a g00( ◦ σa ◦ ¯ σb)AC hlbδCD∂lx h0c hg00δ 3 +hμb(ω μ)CD h0c hg00δ 3 (σ◦ c)D ˙B, (3.5) and SA ˙B(x, y) ←− ∂y0|0= i( ◦ σa)A ˙C h0a hg00δ 3 ←∂−y lδ ˙ C ˙ Dh lb(y) −h 0a hg00(ωμ) ˙ C ˙ Dh μbδ3 (σ¯◦bσ◦c) ˙ D ˙ B h0c(y) g00(y), (3.6)
respectively. These are parallel with (2.26).
We can solve (2.25) in terms of an integral representation,
ξA(x) = d3yJA0(x, y) , (3.7) with J λ A (x, y)≡ i SA ˙B(x, y)h(y)h λa(y)(¯σ◦ a) ˙ BCξ C(y) . (3.8)
Here note that the index λ relates to the vector property at the point y. The bilocal “current”J λ
A (x, y) is conserved with respect to y by virtue of Eqs. (2.24) and (3.4). Therefore, the right-hand side of (3.7) is independent of y0and reduces to ξ
A(x) by setting y0= x0 via (3.2). Inserting (3.7) to the right-hand side of (3.8), we have
ξA(x) = d3z d3y iSA ˙B(x, y)h(y)h0a(y)(σ◦¯a) ˙ BC × i SC ˙D(y, z)h(z)h0b(z)( ◦ ¯ σb)DE˙ ξE(z) . (3.9) Comparing (3.7) and (3.9), we find an integral representation,
SA ˙B(x, z) =
d3y
KA ˙B
with
KA ˙B λ
(x, y, z)≡ i SA ˙C(x, y)h(y)hλa(y)( ◦ ¯ σa) ˙ CD SD ˙B(y, z) . (3.11) The nonlocal “current” KA ˙Bλ(x, y, z) is conserved with respect to y by virtue of Eqs. (2.24), (3.1), and (3.4). Of course, the right-hand side of (3.10) is independent of y0and reduces to
SA ˙B(x, z) by setting y0= x0via (3.2).
Using Eqs. (2.29) and (3.7), we obtain the 4D anti-commutation rela-tion between ξA(x) and (ξ†)B˙(y) as follows:
{ξA(x), (ξ†)B˙(y)} = i SA ˙B(x, y) +RA ˙B(x, y; ξ†) , (3.12)
whereRA ˙B(x, y; ξ†) contains a commutator betweenSA ˙B(x, y) and (ξ†)C˙(z).
4.
Transformation properties
In the quantum coupled Einstein–Weyl system, there are the affine, the gravitational BRST, and the internal Lorentz BRST symmetries. We in-vestigate the transformation properties of the quantum-gravity S function
SA ˙B(x, y) with respect to these three symmetries.
4.1. Affine transformation
Let ˆPλand ˆMκλbe the translation generator and the GL(4) one [1], respec-tively. In what follows, we show that the affine transformation ofSA ˙B(x, y) is given by [ i ˆPλ,SA ˙B(x, y) ] = (∂λx+ ∂ y λ)SA ˙B(x, y) , (4.1) [ i ˆMκλ,SA ˙B(x, y) ] = (x κ∂x λ+ yκ∂ y λ)SA ˙B(x, y) . (4.2) As in Ref. [3], we prove only (4.2) since ˆPλ can formally be regarded as
ˆ
M5
λ with x5= y5= 1, and δ5μ= δν5= 0. Modifying Eqs. (3.1) and (3.4), we see
∂x 0SA ˙B(x, y)|0= i h0a g00( ◦ σa ◦ ¯ σb)AC hlbδCD∂lx h0c hg00δ 3 +hμb(ω μ)CD h0c hg00δ 3 (σ◦ c)D ˙B, (3.5) and SA ˙B(x, y) ←− ∂0y|0= i( ◦ σa)A ˙C h0a hg00δ 3 ←∂−y lδ ˙ C ˙ Dh lb(y) −h 0a hg00(ωμ) ˙ C ˙ Dh μbδ3 (σ◦¯bσ◦c) ˙ D ˙ B h0c(y) g00(y), (3.6)
respectively. These are parallel with (2.26).
We can solve (2.25) in terms of an integral representation,
ξA(x) = d3yJA0(x, y) , (3.7) with J λ A (x, y)≡ i SA ˙B(x, y)h(y)h λa(y)(σ¯◦ a) ˙ BCξ C(y) . (3.8)
Here note that the index λ relates to the vector property at the point y. The bilocal “current”J λ
A (x, y) is conserved with respect to y by virtue of Eqs. (2.24) and (3.4). Therefore, the right-hand side of (3.7) is independent of y0 and reduces to ξ
A(x) by setting y0= x0 via (3.2). Inserting (3.7) to the right-hand side of (3.8), we have
ξA(x) = d3z d3y iSA ˙B(x, y)h(y)h0a(y)(σ¯◦a) ˙ BC × i SC ˙D(y, z)h(z)h0b(z)( ◦ ¯ σb)DE˙ ξE(z) . (3.9) Comparing (3.7) and (3.9), we find an integral representation,
SA ˙B(x, z) =
d3y
KA ˙B
these are derived from the affine transformation of hμa [1],
[i ˆMκλ, hμa] = xκ∂λhμa+ δμκhλa. (4.10) Thus, we find
Aκ
λA ˙B(x, y) = 0 . (4.11)
Hence the set of Eqs. (4.1) and (4.2) is proved. Equation (3.3) and the Hermitian conjugate of (4.11) yield
[ i ˆMκλ,SA ˙B(x, y) ]†=−(x κ∂x
λ+ yκ∂ y
λ)SB ˙A(y, x) . (4.12) On the basis of Eqs. (4.1), (4.2), (4.6), and (4.8), the affine transfor-mation of the integral representation (3.7) for ξA is given by
[i ˆMκ
λ, ξA(x)] = xκ∂λξA(x) +
d3y{∂λy[yκJA0(x, y)]− δ0λJAκ(x, y)} . (4.13) In the right-hand side, the integral term vanish since the “current”JAκ(x, y) is conserved with respect to y.
4.2. Gravitational BRST transformation
We next show that the gravitational BRST transformation ofSA ˙B(x, y) is given by
[ iQG,SA ˙B(x, y) ] =−κ[cρ(x)∂ρxSA ˙B(x, y) +SA ˙B(x, y)←∂− y
ρ· cρ(y)] . (4.14) We define the difference between both sides of (4.14) as follows:
GA ˙B(x, y)≡ [ iQG,SA ˙B(x, y) ] +κ[cρ(x)∂x
ρSA ˙B(x, y) +SA ˙B(x, y)←∂− y
ρ· cρ(y)] . (4.15) We define the difference between both sides of (4.2) as follows:
AκλA ˙B(x, y)≡ [ i ˆM κ λ,SA ˙B(x, y) ]− (x κ∂x λ+ yκ∂ y λ)SA ˙B(x, y) . (4.3) Then we form a Cauchy problem forAκ
λA ˙B(x, y): we apply the operator ihμa ◦σ¯
a (∂μ+ ωμ) and the condition x0= y0 to (4.3). Using Eqs. (2.24), (3.1), (3.2), and (3.4), we obtain i hμa(x)(¯σ◦a)AB˙ {δBC∂μx+ [ωμ(x)]BC}AκλC ˙D(x, y) =−i [ i ˆMκ λ, hμa(x) ]( ◦ ¯ σa) ˙ AB {δ C B ∂μx+ [ωμ(x)]BC}SC ˙D(x, y) −i hμa(x)(σ¯◦ a)AB˙ [ i ˆMκλ, [ωμ(x)]BC]SC ˙D(x, y) +i xκ∂x λhμa(x)· ( ◦ ¯ σa)AB˙ {δBC∂μx+ [ωμ(x)]BC}SC ˙D(x, y) +i xκhμa(x)(σ¯◦ a) ˙ AB∂x λ[ωμ(x)]BC· SC ˙D(x, y) −i hκa(x)(σ¯◦ a)AB˙ ∂λxSB ˙D(x, y) , (4.4) and Aκ λA ˙B(x, y)|0 = i ˆMκ λ,−i h0a hg00( ◦ σa)A ˙Bδ3 +i xκ∂x λ h0a hg00 − δκλ h0a hg00+ δ0 λ hg00 2h 0agκ0 g00 − h κa (σ◦ a)A ˙Bδ 3. (4.5) The right-hand sides of Eqs. (4.4) and (4.5) vanish by virtue of the following commutators: [i ˆMκλ, h] = xκ∂λh + δκλh , (4.6) [i ˆMκλ, gρσ] = xκ∂λgρσ− δλρgκσ− δσλgρκ, (4.7) [i ˆMκλ, hμa] = xκ∂λhμa− δμλh κa, (4.8) [i ˆMκ λ, ωμab] = xκ∂λωμab+ δμκωabλ ; (4.9)
these are derived from the affine transformation of hμa [1],
[i ˆMκλ, hμa] = xκ∂λhμa+ δμκhλa. (4.10) Thus, we find
Aκ
λA ˙B(x, y) = 0 . (4.11)
Hence the set of Eqs. (4.1) and (4.2) is proved. Equation (3.3) and the Hermitian conjugate of (4.11) yield
[ i ˆMκλ,SA ˙B(x, y) ]† =−(x κ∂x
λ+ yκ∂ y
λ)SB ˙A(y, x) . (4.12) On the basis of Eqs. (4.1), (4.2), (4.6), and (4.8), the affine transfor-mation of the integral representation (3.7) for ξA is given by
[i ˆMκ
λ, ξA(x)] = xκ∂λξA(x) +
d3y{∂λy[yκJA0(x, y)]− δλ0JAκ(x, y)} . (4.13) In the right-hand side, the integral term vanish since the “current”JAκ(x, y) is conserved with respect to y.
4.2. Gravitational BRST transformation
We next show that the gravitational BRST transformation ofSA ˙B(x, y) is given by
[ iQG,SA ˙B(x, y) ] =−κ[cρ(x)∂ρxSA ˙B(x, y) +SA ˙B(x, y)←∂− y
ρ · cρ(y)] . (4.14) We define the difference between both sides of (4.14) as follows:
GA ˙B(x, y)≡ [ iQG,SA ˙B(x, y) ] +κ[cρ(x)∂x
ρSA ˙B(x, y) +SA ˙B(x, y)←∂− y
ρ · cρ(y)] . (4.15) We define the difference between both sides of (4.2) as follows:
AκλA ˙B(x, y)≡ [ i ˆM κ λ,SA ˙B(x, y) ]− (x κ∂x λ+ yκ∂ y λ)SA ˙B(x, y) . (4.3) Then we form a Cauchy problem for Aκ
λA ˙B(x, y): we apply the operator ihμa ◦¯σ
a (∂μ+ ωμ) and the condition x0 = y0 to (4.3). Using Eqs. (2.24), (3.1), (3.2), and (3.4), we obtain i hμa(x)(σ¯◦a)AB˙ {δBC∂μx+ [ωμ(x)]BC}AκλC ˙D(x, y) =−i [ i ˆMκ λ, hμa(x) ]( ◦ ¯ σa) ˙ AB {δ C B ∂μx+ [ωμ(x)]BC}SC ˙D(x, y) −i hμa(x)(σ◦¯ a)AB˙ [ i ˆMκλ, [ωμ(x)]BC]SC ˙D(x, y) +i xκ∂x λhμa(x)· ( ◦ ¯ σa)AB˙ {δBC∂μx+ [ωμ(x)]BC}SC ˙D(x, y) +i xκhμa(x)(σ¯◦ a) ˙ AB∂x λ[ωμ(x)]BC· SC ˙D(x, y) −i hκa(x)(σ◦¯ a)AB˙ ∂λxSB ˙D(x, y) , (4.4) and Aκ λA ˙B(x, y)|0 = i ˆMκ λ,−i h0a hg00( ◦ σa)A ˙Bδ3 +i xκ∂x λ h0a hg00 − δκλ h0a hg00 + δ0 λ hg00 2h 0agκ0 g00 − h κa (σ◦ a)A ˙Bδ 3. (4.5) The right-hand sides of Eqs. (4.4) and (4.5) vanish by virtue of the following commutators: [i ˆMκλ, h] = xκ∂λh + δλκh , (4.6) [i ˆMκλ, gρσ] = xκ∂λgρσ− δλρgκσ− δλσgρκ, (4.7) [i ˆMκλ, hμa] = xκ∂λhμa− δλμh κa, (4.8) [i ˆMκ λ, ωabμ ] = xκ∂λωabμ + δκμωλab; (4.9)
Thus, we find
GA ˙B(x, y) = 0 . (4.23)
Hence (4.14) is proved. Equation (3.3) and the Hermitian conjugate of (4.23) yield
[ iQG,SA ˙B(x, y) ]†= κ[cρ(y)∂yρSB ˙A(y, x) +SB ˙A(y, x)←∂−ρx· cρ(x)] . (4.24) On the basis of Eqs. (4.14), (4.18), and (4.20), the gravitational BRST transformation of the integral representation (3.7) for ξAis given by
{iQG, ξA(x)} = −κcρ(x)∂ρxξA(x) +κ
d3y
{∂y
ρ[JA0(x, y)cρ(y)]− JAρ(x, y)∂ρc0(y)} . (4.25) In the right-hand side, the integral term vanish since the “current”JAρ(x, y) is conserved with respect to y.
4.3. Internal Lorentz BRST transformation
In addition, we show that the internal Lorentz transformation ofSA ˙B(x, y) is given by [ iQL,SA ˙B(x, y) ] =−1 2 tab(x)(◦ Sab)ACSC ˙B(x, y)− SA ˙C(x, y)( ◦ Sab) ˙ C ˙ Bt ab(y) . (4.26) We define the difference between both sides of (4.26) as follows:
VA ˙B(x, y)≡ [ iQL,SA ˙B(x, y) ] +1 2 tab(x)(S◦ab)ACSC ˙B(x, y)− SA ˙C(x, y)( ◦ Sab) ˙ C ˙ Bt ab(y) . (4.27) Then we form a Cauchy problem for GA ˙B(x, y): we apply the operator
ihμa ◦σ¯
a (∂μ+ ωμ) and the condition x0= y0 to (4.15). Using Eqs. (2.24), (3.1), (3.2), and (3.4), we obtain i hμa(x)(¯σ◦ a)AB˙ {δBC∂μx+ [ωμ(x)]BC}GC ˙D(x, y) =−i [ iQG, hμa(x) ]( ◦ ¯ σa)AB˙ {δBC∂μx+ [ωμ(x)]BC}SC ˙D(x, y) −i hμa(x)(σ◦¯ a) ˙ AB[ iQ G, [ωμ(x)]BC]SC ˙D(x, y) −iκ cρ(x)∂x ρhμa(x)· ( ◦ ¯ σa)AB˙ {δBC∂μx+ [ωμ(x)]BC}SC ˙D(x, y) −iκ cρ(x)hμa(x)(σ¯◦ a)AB˙ ∂xρ[ωμ(x)]BC· SC ˙D(x, y) +iκ ∂x μcρ(x)· hμa(x)( ◦ ¯ σa) ˙ AB∂x ρSB ˙D(x, y) , (4.16) and GA ˙B(x, y)|0= iQG,−i h0a hg00( ◦ σa)A ˙Bδ 3 −hgiκ00 cρ∂ρh0a− ∂ρc0· hρa −h 0a hg00[∂ρ(c ρhg00) − 2∂ρc0· hg0ρ] (σ◦a)A ˙Bδ 3. (4.17) The right-hand sides of Eqs. (4.16) and (4.17) vanish by virtue of the following commutators:
[ iQG, h ] =−κ∂μ(cμh) , (4.18)
[ iQG, gλμ] = κ(∂νcμ· gλν+ ∂νcλ· gμν− cν∂νgλμ) , (4.19) [ iQG, hμa] = κ(∂νcμ· hνa− cν∂νhμa) , (4.20) [ iQG, ωμab] =−κ(∂μcν· ωνab+ cν∂νωμab) ; (4.21) these are derived from the gravitational BRST transformation of hμa [1],
Thus, we find
GA ˙B(x, y) = 0 . (4.23)
Hence (4.14) is proved. Equation (3.3) and the Hermitian conjugate of (4.23) yield
[ iQG,SA ˙B(x, y) ]†= κ[cρ(y)∂ρySB ˙A(y, x) +SB ˙A(y, x)←∂−xρ· cρ(x)] . (4.24) On the basis of Eqs. (4.14), (4.18), and (4.20), the gravitational BRST transformation of the integral representation (3.7) for ξA is given by
{iQG, ξA(x)} = −κcρ(x)∂ρxξA(x) +κ
d3y
{∂y
ρ[JA0(x, y)cρ(y)]− JAρ(x, y)∂ρc0(y)} . (4.25) In the right-hand side, the integral term vanish since the “current”JAρ(x, y) is conserved with respect to y.
4.3. Internal Lorentz BRST transformation
In addition, we show that the internal Lorentz transformation ofSA ˙B(x, y) is given by [ iQL,SA ˙B(x, y) ] =−1 2 tab(x)(◦ Sab)ACSC ˙B(x, y)− SA ˙C(x, y)( ◦ Sab) ˙ C ˙ Bt ab(y) . (4.26) We define the difference between both sides of (4.26) as follows:
VA ˙B(x, y)≡ [ iQL,SA ˙B(x, y) ] +1 2 tab(x)(S◦ab)ACSC ˙B(x, y)− SA ˙C(x, y)( ◦ Sab) ˙ C ˙ Bt ab(y) . (4.27) Then we form a Cauchy problem for GA ˙B(x, y): we apply the operator
ihμa ◦σ¯
a (∂μ+ ωμ) and the condition x0= y0 to (4.15). Using Eqs. (2.24), (3.1), (3.2), and (3.4), we obtain i hμa(x)(σ¯◦ a)AB˙ {δBC∂μx+ [ωμ(x)]BC}GC ˙D(x, y) =−i [ iQG, hμa(x) ]( ◦ ¯ σa)AB˙ {δBC∂μx+ [ωμ(x)]BC}SC ˙D(x, y) −i hμa(x)(¯σ◦ a) ˙ AB[ iQ G, [ωμ(x)]BC]SC ˙D(x, y) −iκ cρ(x)∂x ρhμa(x)· ( ◦ ¯ σa)AB˙ {δBC∂μx+ [ωμ(x)]BC}SC ˙D(x, y) −iκ cρ(x)hμa(x)(σ◦¯ a)AB˙ ∂ρx[ωμ(x)]BC· SC ˙D(x, y) +iκ ∂x μcρ(x)· hμa(x)( ◦ ¯ σa) ˙ AB∂x ρSB ˙D(x, y) , (4.16) and GA ˙B(x, y)|0= iQG, −i h0a hg00( ◦ σa)A ˙Bδ 3 −hgiκ00 cρ∂ρh0a− ∂ρc0· hρa −h 0a hg00[∂ρ(c ρhg00) − 2∂ρc0· hg0ρ] (σ◦a)A ˙Bδ 3. (4.17) The right-hand sides of Eqs. (4.16) and (4.17) vanish by virtue of the following commutators:
[ iQG, h ] =−κ∂μ(cμh) , (4.18)
[ iQG, gλμ] = κ(∂νcμ· gλν+ ∂νcλ· gμν− cν∂νgλμ) , (4.19) [ iQG, hμa] = κ(∂νcμ· hνa− cν∂νhμa) , (4.20) [ iQG, ωabμ ] =−κ(∂μcν· ωabν + cν∂νωabμ ) ; (4.21) these are derived from the gravitational BRST transformation of hμa [1],
via (2.12).
Applying Eqs. (4.26), (4.30), and (4.31) to the anti-commutation rela-tion between QL and the integral representation (3.7) for ξA, we obtain
{iQL, ξA(x)} = −t ab(x) 2 ( ◦ Sab)ABξB(x) . (4.36)
5.
Discussion
In the present paper, we have briefly reviewed the manifestly covariant operator formalism for Weyl fields in quantum Einstein gravity. We have introduced a quantum-gravity S function SA ˙B(x, y) for Weyl fields as a function satisfying a q-number Cauchy problem given by Eqs. (3.1) and (3.2). Our treatment ofSA ˙B(x, y) has been analogous to that ofS(x, y; m) in Ref. [3]. We have shown that the affine, the gravitational BRST, and the internal Lorentz BRST transformations ofSA ˙B(x, y) are consistent with Eqs. (3.1), (3.2), and (3.7).
As the Weyl field, we have taken ξ(x) rather than η(x) in (2.2). Of course, one can select η(x) with the use ofLη in (2.19), and consequently introduce the quantum-gravity S function ¯SAB˙ (x, y) for it. This S function is defined by the following q-number Cauchy problem:
ih(x)hμa(x)(σ◦a)C ˙D{δ ˙ D ˙ A∂ x μ+ [ωμ(x)] ˙ D ˙ A} ¯S ˙ AB(x, y) = 0 , (5.1) ¯ SAB˙ (x, y)|0=−i h0a(x) h(x)g00(x)( ◦ ¯ σa) ˙ ABδ3. (5.2)
The treatment of ¯SAB˙ (x, y) is parallel with that of S
A ˙B(x, y).
SinceSA ˙B(x, y) is not a function of x− y alone, we must distinguish ∂y from −∂x. On the other hand, if a physical vacuum
|0� is translationally
invariant,
ˆ
Pλ|0� = 0 , (5.3)
Then we form a Cauchy problem for VA ˙B(x, y): we apply the operator
ihμa ◦σ¯
a (∂μ+ ωμ) and the condition x0= y0 to (4.27). Using Eqs. (2.24), (2.37), (3.1), (3.2), and (3.4), we obtain i hμa(x)(σ¯◦ a)AB˙ {δBC∂μx+ [ωμ(x)]BC}VC ˙D(x, y) =−i [ iQL, hμa(x) ]( ◦ ¯ σa)AB˙ {δBC∂μx+ [ωμ(x)]BC}SC ˙D(x, y) −i hμa(x)(σ◦¯ a) ˙ AB[ iQ L, [ωμ(x)]BC]SC ˙D(x, y) +i tabhμ a(x)( ◦ ¯ σb)AB˙ {δBC∂μx+ [ωμ(x)]BC}SC ˙D(x, y) +i 2h μa(x)(σ◦¯ a) ˙ AB[Dx μt(x)]cd( ◦ Scd)BCSC ˙D(x, y) , (4.28) and VA ˙B(x, y)|0= iQL,−i h 0a hg00( ◦ σa)A ˙Bδ3 + itabh 0 a hg00( ◦ σb)A ˙Bδ3. (4.29) The right-hand sides of Eqs. (4.28) and (4.29) vanish by virtue of the following commutators:
[ iQL, h ] = 0 , (4.30)
[ iQL, hμa] =−tabhμb, (4.31) [ iQL, ωμab] = (Dμt)ab; (4.32) these are derived from the internal Lorentz BRST transformation of hμa [1],
[ iQL, hμa] =−tabhμb. (4.33) Thus, we find
VA ˙B(x, y) = 0 . (4.34)
Hence (4.26) is proved. Equation (3.3) and the Hermitian conjugate of (4.34) yield
[ iQL,SA ˙B(x, y) ]† = 1
2
tab(y)(S◦ab)BCSC ˙A(y, x)− SB ˙C(y, x)( ◦ Sab) ˙ C ˙ At ab(x) , (4.35)
via (2.12).
Applying Eqs. (4.26), (4.30), and (4.31) to the anti-commutation rela-tion between QL and the integral representation (3.7) for ξA, we obtain
{iQL, ξA(x)} = −t ab(x) 2 ( ◦ Sab)ABξB(x) . (4.36)
5.
Discussion
In the present paper, we have briefly reviewed the manifestly covariant operator formalism for Weyl fields in quantum Einstein gravity. We have introduced a quantum-gravity S function SA ˙B(x, y) for Weyl fields as a function satisfying a q-number Cauchy problem given by Eqs. (3.1) and (3.2). Our treatment ofSA ˙B(x, y) has been analogous to that ofS(x, y; m) in Ref. [3]. We have shown that the affine, the gravitational BRST, and the internal Lorentz BRST transformations ofSA ˙B(x, y) are consistent with Eqs. (3.1), (3.2), and (3.7).
As the Weyl field, we have taken ξ(x) rather than η(x) in (2.2). Of course, one can select η(x) with the use ofLη in (2.19), and consequently introduce the quantum-gravity S function ¯SAB˙ (x, y) for it. This S function is defined by the following q-number Cauchy problem:
ih(x)hμa(x)(σ◦a)C ˙D{δ ˙ D ˙ A∂ x μ+ [ωμ(x)] ˙ D ˙ A} ¯S ˙ AB(x, y) = 0 , (5.1) ¯ SAB˙ (x, y)|0=−i h0a(x) h(x)g00(x)( ◦ ¯ σa) ˙ ABδ3. (5.2)
The treatment of ¯SAB˙ (x, y) is parallel with that ofS
A ˙B(x, y).
SinceSA ˙B(x, y) is not a function of x− y alone, we must distinguish ∂y from−∂x. On the other hand, if a physical vacuum
|0� is translationally
invariant,
ˆ
Pλ|0� = 0 , (5.3)
Then we form a Cauchy problem for VA ˙B(x, y): we apply the operator
ihμa ◦σ¯
a (∂μ+ ωμ) and the condition x0= y0 to (4.27). Using Eqs. (2.24), (2.37), (3.1), (3.2), and (3.4), we obtain i hμa(x)(σ¯◦ a)AB˙ {δBC∂μx+ [ωμ(x)]BC}VC ˙D(x, y) =−i [ iQL, hμa(x) ]( ◦ ¯ σa)AB˙ {δBC∂μx+ [ωμ(x)]BC}SC ˙D(x, y) −i hμa(x)(¯σ◦ a) ˙ AB[ iQ L, [ωμ(x)]BC]SC ˙D(x, y) +i tabhμ a(x)( ◦ ¯ σb)AB˙ {δBC∂μx+ [ωμ(x)]BC}SC ˙D(x, y) +i 2h μa(x)(σ¯◦ a) ˙ AB[Dx μt(x)]cd( ◦ Scd)BCSC ˙D(x, y) , (4.28) and VA ˙B(x, y)|0= iQL,−i h 0a hg00( ◦ σa)A ˙Bδ3 + itabh 0 a hg00( ◦ σb)A ˙Bδ3. (4.29) The right-hand sides of Eqs. (4.28) and (4.29) vanish by virtue of the following commutators:
[ iQL, h ] = 0 , (4.30)
[ iQL, hμa] =−tabhμb, (4.31) [ iQL, ωabμ ] = (Dμt)ab; (4.32) these are derived from the internal Lorentz BRST transformation of hμa [1],
[ iQL, hμa] =−tabhμb. (4.33) Thus, we find
VA ˙B(x, y) = 0 . (4.34)
Hence (4.26) is proved. Equation (3.3) and the Hermitian conjugate of (4.34) yield
[ iQL,SA ˙B(x, y) ]† =1
2
tab(y)(S◦ab)BCSC ˙A(y, x)− SB ˙C(y, x)( ◦ Sab) ˙ C ˙ At ab(x) , (4.35)
then the vacuum expectation value of (4.1) is given by
�0| [ i ˆPλ,SA ˙B(x, y) ]|0� = (∂λx+ ∂ y
λ)�0|SA ˙B(x, y)|0� = 0 , (5.4) These mean that ∂λy is equivalent to−∂x
λ. References
[1] N. Nakanishi and I. Ojima, Covariant Operator Formalism of Gauge Theories and Quantum Gravity (World Scientific, Singapore, 1990). [2] M. Abe and N. Nakanishi, Prog. Theor. Phys. 88, 975 (1992). [3] R. Yoshida, Prog. Theor. Exp. Phys. 2014, 103B06 (2014).
[4] L. H. Ryder, Quantum Field Theory, 2nd Edition (Cambridge Uni-versity Press, Cambridge, 1996).
